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Math Help - Find two positive divergent sequences such that..

  1. #1
    Super Member General's Avatar
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    Find two positive divergent sequences such that..

    Hello
    Find a_n and b_n such that:
    { an} , { b_n} are divergent sequences
    a_n,b_n > 0 for n\geq1
    and { a_n+b_n } is a convergent sequences.

    The problem here is both must positive.
    If not, a_n=n and b_n=-n is a nice counter-example.

    But they must be positive.
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  2. #2
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    Quote Originally Posted by General View Post
    Hello
    Find a_n and b_n such that:
    { an} , { b_n} are divergent sequences
    a_n,b_n > 0 for n\geq1
    and { a_n+b_n } is a convergent sequences.

    The problem here is both must positive.
    If not, a_n=n and b_n=-n is a nice counter-example.

    But they must be positive.

    Here's a tip, just because a sequence is divergent it doesn't mean it goes to infinity.

    Bigger hint;
    Spoiler:
    think sine and cosine.
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    Quote Originally Posted by pomp View Post
    Here's a tip, just because a sequence is divergent it doesn't mean it goes to infinity.

    Bigger hint;
    Spoiler:
    think sine and cosine.
    But sine and cosine are not positive for all n\geq1
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    Quote Originally Posted by General View Post
    But sine and cosine are not positive for all n\geq1
    Well then do something to them that will ensure they are...
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  5. #5
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    You mean sin^2n and cos^2n ?
    What the limit of them ?
    What the limit of thier sum ?
    as n \rightarrow \infty
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    Quote Originally Posted by General View Post
    You mean sin^2n and cos^2n ?
    What the limit of them ?
    What the limit of thier sum ?
    as n \rightarrow \infty
    "What the limit of them?"

    Neither converge to a limit, they oscillate in (0,1)

    "What is the limit of thier sum? "

    What is \sin^2 (x) + \cos^2(x) ?
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